本文我們考慮在平衡單因子常態隨機效應模型下去建構一個雙邊的β-content容許界限，其方法是根據Chen and Harris (2006)限制於未知期望均方比率的估計量所推導出。本文並探討比較Mee(1984)、Beckman 和Tietjen (1989)及Liao 和 Iyer(2004)所提及的雙邊容許界限，透過蒙地卡羅模擬方法去計算這四種方法的雙邊容許界限的覆蓋率、平均值和標準差；及引用三個實際例子，比較這四種方法的雙邊容許界限的表現。模擬方法研究表示本文的方法比目前存在的方法較不保守且有較準確的覆蓋率。本文亦提供所需要的統計表格。

We consider in this article a method of constructing a two-sided β-content tolerance limits for the balanced one-way normal random effects model. The technique is derived by conditioning on an estimator of the unknown expected mean square ratio as that in Chen and Harris (2006). We also compare the present procedure with the tolerance limits of Mee (1984), Beckman and Tietjen (1989), and Liao and Iyer (2004). In addition, we include three examples from Lemon (1977) and Vangel (1992). Simulation studies indicate that the present procedure is less conservative than several current existing methods and gives more accurate coverage rates. Statistical tables needed to implement the procedure are also included.

1. 前言 1
2.文獻回顧 2
2.1 單因子隨機效應模型下的容許界限 2
2.2 目前常用的雙邊容許界限 3
2.2.1 Mee (1984) 3
2.2.2 Beckman 和 Tietjen (1989) 4
2.2.3 Liao 和 Iyer (2004) 5
2.3 本文雙邊容許界限 7
3.蒙地卡羅模擬研究 10
4.實例比較 21
4.1 實例一 21
4.2 實例二 22
4.3 實例三 23
5.結論 25
參考文獻 26
附錄
Fortran程式 34
R程式 38 

表目錄
表2-1:單因子隨機效應模型的變異數分析表 2
表 3.1 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 11
表 3.2 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 13
表 3.3 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 14
表 3.4 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 16
表 3.5 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 17
表 3.6 P=.90 和 γ=.95, 蒙地卡羅模擬估計
       四種雙邊容許界限的覆蓋率,平均值和標準差 19
表 4-1 實例二: 五批材料抗拉力的觀測值 22
表 4-2 實例三: 五批材料抗拉力的觀測值 23
表 A.1 P=.95 和 γ=.90, 雙邊容許界限的 η2 值  28
表 A.2 P=.95 和 γ=.95, 雙邊容許界限的 η2 值  29
表 A.3 P=.95 和 γ=.99, 雙邊容許界限的 η2 值  30
表 A.4 P=.99 和 γ=.90, 雙邊容許界限的 η2 值  31
表 A.5 P=.99 和 γ=.95, 雙邊容許界限的 η2 值  32
表 A.6 P=.99 和 γ=.99, 雙邊容許界限的 η2 值  33

1. 蔡雅屏 (2004) 台北大學統計學系碩士論文，``多個容許界限在單因子隨機效應模型下的表現.``
2. Beckman, R. J. and Tietjen, G. L. (1989).Two-Sided Tolerance Limits for Balanced Random-Effects ANOVA Models. Technometrics 31, 185-197.
3. Beckman, R. J. and Tietjen, G. L. (1998).Correction: Two-Sided Tolerance Limits for Balanced Random-Effects ANOVA Models. Technometrics 40, 269.
4. Chen, S. Y. and Harris, B. (2006).On Lower Tolerance Limits With Accurate Coverage Probabilities for the Normal Random Effects Model. Journal of the American Statistical Association 101, 1039-1045.
5. Jordan, P. (1995).Estimation of Tolerance Limits from Reference Data.Computational Statistics & Data Analysis 19, 655-668.
6. Lemon, G. H. (1977).Factors for One-Sided Tolerance Limits for Balanced One-Way-ANOVA Random-Effects Model. Journal of the American Statistical Association 72, 676-680.
7. Liao, C. T. and Iyer, H. K. (2004).A Tolerance Interval for the Normal Distribution With Serveval Variance Components. Statistica Sinica 14, 217-229.
8. Lin, T. Y. and Liao, C. T. (2006).A β-Expectation Tolerance Interval for General Balanced Mixed Linear Models. Computational Statistics & Data Analysis 31,911-925.
9. Mee, R. W. (1984).β-Expectation and β-Content Tolerance Limits for Balanced One-Way ANOVA Random Model. Technometrics 26, 251-254.
10. Satterthwaite, F. E. (1946).An Approximate Distribution of Estimate of Variance Components. Biometrics Bulletin 2, 110-114.
11. Vangel, M. G. (1992). New Methods for One-Sided Tolerance Limits for a One-Way Balanced Random-Effects ANOVA Model. Technometrics 34, 176-185.
12. Weerahandi, S.(1993).Generalized Confidence Intervals. Journal of the American Statistical Association 88, 899-905.